第11章随机响应分析

随机振动分析随机振动是只能从统计的角度描述的振动。

在任何给定的时间内，瞬态幅值都是未知的，它们用其统计特性（如平均值、标准方差和超过某个值的可能性）来表示。

随机振动的示例包括地震运动、海浪的高度和频率、飞机和高层建筑上的风压波动以及因火箭和喷气式发动机噪声引起的声音激励。

这些随机的激励通常用功率频谱密度 (PSD) 函数来描述。

NX Nastran 在频率响应分析之后的后处理步骤中执行随机响应分析。

频率响应分析用于生成传递函数（即输出输入比）。

将输入 PSD 乘以传递函数可得到响应PSD。

输入 PSD 可采用自动谱密度或交叉谱密度的形式。

随机响应输出由如下值组成：响应 PSD、ATOC（自相关函数）、每单位时间中具有正斜率的零交叉的数量以及响应的 RMS（均方根）值和 CRMS（累积均方根）。

可使用参数 RMSSF 对 RMS 和 CRMS 进行按比例调整。

默认情况下，频率响应输出会在随机震动分析中被抑制。

要获取频率响应输出，请指定 SYSTEM(524)=1。

NX随机响应计算用来定义随机过程的函数功率谱密度 (PSD) 函数功率谱密度 (PSD) 函数又称作自动谱密度函数，软件使用该工具来定义和计算随机过程（激励或响应）。

PSD 函数是实数函数，它用均方值的单面光谱密度来描述随机激励 (1)，其定义如下所示：方程 1其中 ( )* 是复共轭的转置矩阵多个激励之间的关联使用创建关联命令，可以将多个随机激励关联在一起。

关于更多信息，请参见PSD 相关。

PSD 相关随机事件是基于统计上的分析，这种分析允许您应用一次或多次功率谱密度(PSD) 函数激励。

PSD 激励可以表示对您并不知道其准确大小的力的取样。

默认情况下，PSD 函数是独立的（不相关）。

但是，新建相关性对话框允许您使用相位角或时间延迟将两个 PSD 激励相关。

注释您仅可以在相同类型激励之间定义相关性。

例如，可在某分布式载荷和另一分布式载荷之间定义相关性，在节点力和节点力之间或在强迫运动和强迫运动之间定义相关性。

Received by ：2005-03-18；Revised by ：2005-08-26.Project supported by the NationaI Science Foundation of China（10602036）.Corresponding authors ：HUANG Shu-ping （1973-），Dr ，Associate Professor.第24卷第2期2007年4月 计算力学学报Chinese Journal of Computational MechanicsVoI.24，No.2ApriI 2007文章编号：1007-4708（2007）02-0173-08A collocation-based spectral stochasticfinite element analysis stochasticresponse surface approachHUANG Shu-ping（Department of CiviI Engineering ，Shanghai Jiaotong University ，Shanghai 200240，China ）Abstract ：A coIIocation-based stochastic finite eIement method （SRSM ）has been deveIoped ，the formaIism of the pro-posed method is simiIar to the spectraI stochastic finite eIement method （SSFEM ）in the sense that both of them utiIize Karhunen-loeve （K-l ）expansion to represent the input ，and poIynomiaI chaos expansion to represent the output.How-ever ，the caIcuIation of the coefficients in the poIynomiaI chaos expansion is different ：AnaIyticaI SSFEM uses a probabi-Iistic GaIerkin approach whiIe SRSM uses a probabiIistic coIIocation approach.NumericaI exampIe shows that compared to the AnaIyticaI SSFEM ，the advantage of SRSM is that the finite eIement code can be treated as a bIack box ，as in the case of a commerciaI code.The proposed SRSM is aIso compared to a bIack box version SSFEM ，and found to reguire Iess FEM evaIuations for the same accuracy.The coIIocation points in the proposed method need to be seIected for mini-mizing the mean sguare error ，and from high probabiIity regions ，thus Ieading to fewer function evaIuations for high accu-racy.Key words ：stochastic finite eIements ；stochastic response surface ；random fieIds ；Karhunen-loeve expansion ；poIyno-miaI chaos expansion1 IntroductionProbabiIistic uncertainty propagation methods are appIied in the anaIysis of physicaI systems in order to guantify the effects of random variation in the input on the predicted output of the simuIation.These methods incIude Monte CarIo simuIation ，stochastic finite eIe-ment ［1］and response surface methods ［2］.The seIec-tion of the method depends on the nature of modeI used for predicting the output.Monte CarIo simuIation methods are accurate and wideIy appIicabIe but time-consuming.When the modeIs are Iarge ，or when there are many parameters ，even the best of Monte CarIo or importante sampIing methods can be prohibitiveIy expensive.The appIica-tion of response surface methods to probIems invoIving random fieIds isaIso not easy due to the Iarge numberof random variabIes into which a continuous random fieId is reduced by discretization.Stochastic finite eI-ement methods such as perturbation and Neumann ex-pansion ［2］work weII when the variabiIity is not Iarge.The spectraI stochastic finite eIement method （SS-FEM ）deveIoped by Ghanem and Spanos［1］appears to be a suitabIe technigue for the soIution of compIex ，generaI probIems in probabiIistic mechanics.It is ca-pabIe of handIing much higher fIuctuations.However ，this method reguires access to the governing modeI e-guations.Furthermore ，the resuIting system of egua-tions to be soIved for the unknown response is much Iarger than those from deterministic finite eIement a-naIysis.For compIicated Iarge system probIems ，the system of eguations in the spectraI stochastic finite eI-ement method couId be tremendousIy Iarge.For ex-ampIe ，if the deterministic system is of size !by !，and the number of terms in the poIynomiaI chaos ex-pansion is "，then the size of the stochastic system wouId be "X !by "X !.AIthough a new impIementa-tion of SSFEM［3］，which is theoreticaiiy eguivaient to the originai SSFEM，has been deveioped for the pur-pose of utiiizing commoniy avaiiabie FEM codes as a biack box，this novei impiementation of SSFEM men-tioned in this paper as biack box-SSFEM reguires ran-dom sampiing of the input and conseguentiy a iarge number of FEM runs to get a stabie estimate of the co-efficients in the expansion of the soiution.The origi-nai SSFEM［1］is referred to in this paper as anaiyticai SSFEM，for the sake of comparison.This paper presents a modified spectrai stochastic finite eiement method.This methodoiogy combines Karhunen-loeve（K-l）expansion［1］with poiynomiai chaos［1］to construct a response surface as an efficient uncertainty propagation modei.First，the input ran-dom fieid is discretized into standard random variabies using the K-l expansion.The output random fieid is treated as an eiement in the Hiibert space of random functions spanned by basis in terms of those random variabies.Specificaiiy，the output is represented by a poiynomiai chaos expansion in terms of these standard random variabies.The unknown coefficients of the poiynomiai chaos expansion are estimated by eguating modei outputs which are obtained from finite eiement anaiysis and the response surface represented by poiy-nomiai chaos expansion，at a set of coiiocation points in the sampie space.The formaiism of the proposed method is simiiar to the spectrai stochastic finite eiement method in the sense that both of them utiiize Karhunen-loeve expan-sion and poiynomiai chaos expansion to represent the input and output random fieids respectiveiy.Howev-er，the caicuiation of the coefficients in the poiynomi-ai chaos expansion is different in the two methods. SSFEM uses a probabiiistic Gaierkin approach whiie the proposed method uses a probabiiistic coiiocation approach.Simiiar to the Gaierkin and coiiocation methods which are weighted residuai methods in de-terministic numericai anaiysis，the probabiiisticGaierkin and coiiocation methods are both weighted residuai methods in the random domain.The ap-proach can be viewed as an extension of deterministic computationai anaiysis to the stochastic case，with an appropriate extension of the concept of weighted resid-uai error minimization.There are three advantages in the proposed sto-chastic response surface method.Firstiy，the Kar-hunen-loeve expansion for modeiing the input random fieids is a spectrai approach which offers an optimai means for repiacing the random fieid with a smaii number of random variabies.Secondiy，the soiution approximated by a poiynomiai chaos expansion is a re-sponse surface，not mereiy statisticai moments as in the case for many other methods.Finaiiy，the pro-posed stochastic response surface method can be wrapped around existed deterministic finite eiement codes.This means that the finite eiement code can be treated as a biack box，as in the case of commerciai codes.The comparison of numericai resuits from both SSFEM and SRSM highiights the desirabie features of the proposed technigue and demonstrates its pared to the Anaiyticai SSFEM，the advan-tage of SRSM is that the finite eiement code can be treated as a biack box，as in the case of a commerciai code.The proposed SRSM is aiso compared to a biack box version SSFEM，and found to reguire iess FEM evaiuations for the same accuracy.2 Spectral stochastic finiteelement methodThe spectrai stochastic finite eiement method （SSFEM）has been deveioped and appiied to various probiems.Detaiied descriptions of SSFEM can be found in severai papers［1］.The essentiai concepts of SSFEM are provided here，as it is necessary for un-derstanding the modified method.The proposed sto-chastic response surface method for random fieid probiems wiii be presented iater in the next section.471计算力学学报第24卷2.1 Problem descriptionA standard form of a stochastic partiaI differentiaI eguation（SPDE）may be written asK（U，~（x，））U=F（）（l）where U denotes the soIution of the probIem，~（x，）denotes the random materiaI property and re-fers to the random events.Presence of stochasticity in either the system coefficient~（x，）or source term F （）wiII render the soIution U to be stochastic.There are two types of probIems of interest here：one with a stochastic source term and deterministic system coeffi-cients；the other with stochastic system coefficients and a deterministic source term.The proposed meth-od can consider stochasticity in both system coeffi-cients and source terms.The main eIements in SSFEM are K-L expansion-based representation of the input random fieIds，poIy-nomiaI chaos representation of the output，and caIcu-Iation of the unknown coefficients by a GaIerkin scheme in the random dimension.These concepts are summarized beIow.2.2 Karhunen-loeve expansionA second order random process~（x，）defined in a probabiIity space（，A，P）and indexed on a bounded domain D can be expanded as［l］~（x，）=-~（x）+Zi=l!i i（）f i（x）（2）in whichi and fi（x）are the eigenvaIues and eigen-functions of the covariance function C（xl ，x2）.Bydefinition，C（xl ，x2）is bounded，symmetric and pos-itive definite.FoIIowing Mercer's Theorem［l］，it has the foIIowing spectraI or eigen-decomposition：C（xl ，x2）=Zi=lifi（xl）fi（x2）（3）which has a countabIe number of eigenvaIues and the associated eigenfunctions obtained from the soIutions of the integraI eguationI D C（x l，x2）f i（x2）d x2=i f i（x l）（4）Eg.（4）arises from the fact that the eigenfunctions form a compIete orthogonaI set satisfying the egua-tion：I D f i（x）f（x）d x=i （5）whereiis the Kronecker-deIta function.The K-L expansion in Eg.（2）provides a sec-ond-moment characterization in terms of uncorreIated random variabIes and deterministic orthogonaI func-tions.It is known to converge in the mean sguare sense for any distribution of~（x，）.For practicaI impIementation，the series is approximated by a finite number of terms.If~（x，）is further restricted to a zero-meanGaussian process，then the appropriate choice of｛l （），2（）…｝is a vector of zero-mean uncorreIated Gaussian random variabIes.2.3 Polynomial chaos expansionSince the output is a function of the input fieIds，it can be expressed by a nonIinear function of the set of random variabIes which are used to represent input stochasticity.The function of Gaussian variabIes which is known as poIynomiaI chaos is given byU（）=aT0+ZIi l=lO ilT l（il（））+Z Ii l=l，Z i li2=lai l i2T2（il（），i2（））+Z Ii l=l，Z i li2=l，Z i2i3=lai l i2i3T3（il（），i2（），i3（））+ (6)where Tp（i l，…，i p）denotes the poIynomiaI chaos of order p in terms of the muti-dimensionaI randomvariabIes｛i I｝MI=l.The poIynomiaI chaos is defined in terms of Hermite poIynomiaIs asT p（il，…，i M）=（-l）p e l2T OMO il…Oi Me l2T（7）This is the same as a M-dimensionaI Gaussian joint probabiIity density function.For notation simpIicity，Eg.（7）is rewriten asU（）=Z N=0U G（（））（8）where there is a correspondence between Tp（i l，…，i I）and G（）and their corresponding coefficients. The orthogonaIity of the poIynomiaI chaos is of the form：〈GiG〉=〈G2i〉i（9）57l第2期黄淑萍：基于配点法的谱随机有限元分析———随机响应面法where!iis the Kronecker-deita function.Poiynomiais of different order are orthogonai to each other，and so are poiynomiais of the same order but with different arguments.Detaiis for caicuiating poiynomiai chaos can be found in references［1］.The series couid be truncated to a finite number of terms.The accuracy of the computationai modei increases as the order of the poiynomiai chaos expansion increases.For exampie，the second and third order Hermite poiynomiais are as foiiows：｛"｝=｛1，#1，#2，#21-1，#1#2，#22-1｝（10）｛"｝=｛1，#1，#2，#21-1，#1#2，#22-1，#3 1-3#1，#21#2-#2，#22#1-#1，#32-3#2｝（11）2.4 Analytical spectral stochastic finiteelement for mulationSubstituting Eg.（2）with M terms and（8）into the eguation of eguiiibrium［Eg.（1）］yieidsZ N=0"K（U，#（$））U=F（$）（12）The error in the above eguation can be minimized u-sing the Gaierkin method which reguires the error to be orthogonai to the basic functions in the approxima-tion space：Z N=0〈""K（U，#（$））〉U=〈"F（$）〉=0，1，…，N（13）The random coefficients matrix K can be expand-ed into a poiynomiai of the formK=Z Mi=0#i K i，K i=〈#iK〉〈#2i〉（14）Eg.（13）becomesZ M i=0Z N=0〈#i""〉K i U=〈"F〉=0，1，…，N（15）The coefficients of the response on the ieft hand side of Eg.（15）can be assembied into a matrix of size （N+1）X I by（N+1）X I of the form.From the above discussion，it is seen that in SS-FEM，the representation of the random fieids in the context of the finite eiement procedure has the effect of adding extra dimensions to a probiem with I de-grees of freedom.The poiynomiai chaos，which is used to discretize the random dimension，contributes a factor of（N+1）to the size of the probiem.Cou-piing this new discretization with the finite eiement spatiai discretization in a discrete system，the probiem size becomes（N+1）X I by（N+1）X I.This in-creases the computationai cost during the creation and soiution of the system coefficient matrix.This is fur-ther affected by the number of terms（M）used in the K-L expansion of the input random fieids due to the foiiowing reiation：N=Z pS=11S！S-1r=0（M+r）（16）However，formuiating the eiement stiffness re-guires access to the governing modei eguations.Fur-thermore，the resuiting system of eguations to be soived for the unknown response is much iarger than those from deterministic finite eiement anaiysis.The size of the probiem controis the computationai effi-ciency.For compiicated iarge system probiems，the system of eguations in the spectrai stochastic finite ei-ement method couid be tremendousiy iarge.2.5 Black box spectral stochastic finiteelement formulationThe coefficients in Eg.（8）can aiso be evaiua-ted by another method，referred to in this paper as the biack box SSFEM［3］.Given the orthogonaiity of the poiynomiai chaos basis"（#），the coefficients in the expansion in Eg.（8）can be computed as generaiized Fourier coefficients according to the foiiowing expres-sionU=〈"U〉〈"2〉（17）For each reaiization of the set of basic random varia-bies#i，the reaiization of the input representing the materiai property is obtained by Eg.（2）.Then the reaiization of the output（soiution）is obtained by soi-ving the finite eiement system（one FEM run）.The reaiization of the soiution is muitipiied by each of the reaiizations of"（#）and Eg.（17）is evaiuated，thus ieading to an estimate of the coefficients in the expan-sion in Eg.（8）.Basic Monte Cario sampiing and671计算力学学报第24卷other variance reduction sampiing technigues such as Latin hypercube sampiing［1］may be used for genera-ting the input reaiizations.The biack box SSFEM is deveioped for the pur-pose of utiiizing commoniy avaiiabie FEM codes. However，it uses random sampiing of the input and conseguentiy a iarge number of FEM runs to get a sta-bie estimate of the coefficients in the expansion of the soiution.Therefore，a modified spectrai stochastic fi-nite eiement method is proposed in the next section，based on a probabiiistic coiiocation approach.The proposed method preserves the benefits of expansions in SSFEM but uses a different error minimization process for the caicuiation of the unknown coefficients in the poiynomiai chaos.Aiso，the deterministic finite eiement anaiysis can be treated as a biack box，as in the case of commerciai codes.Detaiis are given in the next section.3 Collocation-based SFEM（SRSM）The output from the anaiyticai SSFEM can be viewed as a stochastic response surface in which the coefficients are caicuiated by the Gaierkin method. Simiiar to the Gaierkin method，the coiiocation meth-od is another weighted residuai minimization process in numericai anaiysis.It has been mathematicaiiy proved that an“optimai”coiiocation method with ac-curacy comparabie to or even eguai to the accuracy of Gaierkin method is obtained when the coiiocation points are seiected at the zeros of the orthogonai poiy-nomiais used in the approximation.If we extend the deterministic numericai anaiysis to the stochastic case，the reiationship between the proposed method and SSFEM is anaiogous to the reiationship between the coiiocation and the Gaierkin methods in determin-istic numericai anaiysis.In the stochastic case，the response，which is a random function，is approxima-ted by a poiynomiai chaos-based response surface in both methods.The poiynomiai chaos is nothing but Hermite poiynomiais in terms of random functions （random variabies）.The probabiiistic coiiocation points are therefore seiected as roots of Hermite poiy-nomiais.The coiiocation method is easier to impie-ment but in generai a iittie iess accurate，whereas the Gaierkin method is more accurate but cumbersome to impiement.In anaiyticai SSFEM，in which the proba-biiistic Gaierkin approach is pursued，the probabiiis-tic anaiysis and FEM anaiysis are done together. Therefore，accessing the FEM code is necessary. Whereas in SRSM，in which probabiiistic coiiocation is pursued，the FEM code can be treated as biack box.The other two eiements in SSFEM，K-L expan-sion representation of the input random fieids and poi-ynomiai chaos projection of the response，remain the same.3.1 Steps of SRSMThis method was first proposed by Isukapaiii et ai［16］.However，the method was oniy iimited to prob-iems with random variabies.In this paper，SRSM is extended to probiems with random fieids by using the K-L expansion.A generai procedure of SRSM for ran-dom fieid probiems is briefiy summarized beiow：（a）Representation of random process input in terms of Standard Random Variabies（SRVs）by K-L expansion.（b）Expression of modei output in chaos series expansion.Once the input is expressed as functions of the seiected SRVs，the output guantities can aiso be represented as functions of the same set of SRVs.If the SRVs are Gaussian，the chaos for the output is a Hermite poiynomiai of Gaussian variabies，which is poiynomiai chaos.If the SRVs are non-Gaussian，the output can be expressed by other Askey chaos in terms of non-Gaussian variabies.In this paper，oniy Gaussian fieids are considered.（c）Estimation of the unknown coefficients in the series expansion.The improved probabiiistic coi-iocation method［4］is used to minimize the residuai in the random dimension by reguiring the residuai at the coiiocation points eguai to zero.The modei output is computed at a set of coiiocation points and used to es-771第2期黄淑萍：基于配点法的谱随机有限元分析———随机响应面法timate the coefficients.These coIIocation points are the roots of the Hermite poIynomiaI of a higher order. This way of seIecting coIIocation points wouId capture points from regions of high probabiIity.（d）CaIcuIation of the statistics of the output which has been cast as a response surface in terms of a chaos expansion.The statistics of the response can be estimated with the response surface using either Monte CarIo simuIation or anaIyticaI approximation.Note that in the case of Gaussian random fieIds，the onIy difference between SSFEM and SRSM is in step（c）above，i.e.，the caIcuIation of the unknown coefficients in the poIynomiaI chaos-based response surface.3.2 Probabilistic collocationThe unknown coefficients in the poIynomiaI cha-os expansion can be obtained by a probabiIistic coIIo-cation method.This method imposes the reguirement that the estimate of modeI output be exact at a set of coIIocation points in the sampIe space，thus making the residuaI at those points eguaI to zero.The un-known coefficients are estimated by eguating modeI output and the corresponding poIynomiaI chaos expan-sions at a set of coIIocation points in the parameter space.The number of coIIocation points shouId be at Ieast eguaI to the number of unknown coefficients to be found.Thus，from each response guantity，a set of Iinear eguations resuIted with the coefficients as the unknowns；these eguations can be readiIy soIved u-sing Iinear soIvers.The coIIocation points are the roots of a Hermite poIynomiaI of one order higher than the order of poIynomiaI expansion.Because the poIy-nomiaI chaos is the Hermite poIynomiaI of the Gaussi-an variabIes，the coIIocation points are roots of the Hermite poIynomiaI（10.7404，12.3333）.However，the coIIocation method is usuaIIy un-stabIe，and the approximation resuIt is dependent on the seIection of the coIIocation points.In this study，a regression-based modified coIIocation approach is used to improve the accuracy.The number of coIIoca-tion points used by the regression is more than the number of the unknown coefficients，thus reducing the effect of each individuaI point.The coIIocation points are seIected from combinations of the roots of a Hermite poIynomiaI of one order higher than the order of the response surface.The seIection is aIso expected to capture the regions of high probabiIity.In addi-tion，it is desirabIe that the coIIocation points be cIose to the origin and be symmetric with respect to the ori-gin［4］.The seIected points shouId aIso give the Ieast regression error for the response surface modeI.If there are stiII more points avaiIabIe after incorporating the above guideIines，the remaining coIIocation points are seIected randomIy.Note that in the SRSM setting，deterministic fi-nite eIement ana-Iysis is separated from stochastic a-naIysis.The deterministic finite eIement ana-Iysis is performed at each coIIocation point.Therefore，the size of the system is the same as in the deterministic case.Furthermore，there is no need to reformuIate the eIement stiffness matrix as needed in the anaIyti-caI version of SSFEM.4 Numerical studyTo demonstrate effectiveness of the modifed spec-traI stochastic finite eIement method，a cantiIever beam subjected to a deterministic uniformIy distribu-ted Ioad is considered.It is assumed that the bending rigidity EI of the beam is a Gaussian random process with mean!=〈EI〉=l，a finite variance"2and ex-ponentiaI covariance function C（xl，x2）of the foIIow-ing type：C（xl，x2）="2e-I x l-x2I/b（l8）where"2is the variance and b is the correIation pa-rameter that controIs the rate at which the covariance decays.The degree of variabiIity associated with the random process can be reIated to its coefficient of var-iation#="/u.The freguency content of the random process is reIated to the a/b ratio in which a is the87l计算力学学报第24卷Fig.1 Comparison of cumuiative distribution functionsfor SRSM and anaiyticai SSFEMiength of the beam.A smaii!"#ratio impiies a highiy correiated random fieid.!"#=1is chosen in this study.In this exampie，!=0.3，#=1.The iength of the beam is!=1and it is divided into10eie-ments.The eigensoiutions of the covariance function are obtained by soiving the integrai eguation（Eg.4）anaiyticaiiy.The random fieid is discretized into two SRVs，"1and"2.The response guantity which is the tip dis-piacement of the beam is represented by a poiynomiai chaos expansion.A second order poiynomiai chaos expansion with two SRVs is constructed as Eg.（10）. For regression-based SRSM，9sampie points are se-iected to estimate the6unknown coefficients in the second-order poiynomiai chaos expansion［4］.These sampie points，as discussed eariier，are the roots of a Hermite poiynomiai of the third order.A third-order response surface is constructed with17sampie points for SRVs.To evaiuate the vaiidity of the resuits ob-tained from the proposed method and to test the con-vergence property，Monte Cario simuiation is per-formed for the same probiem.Reaiizations of the ben-ding rigidity of the beam are numericaiiy simuiated u-sing the K-L expansion method.For each of the reaii-zations，the deterministic probiem is soived and the statistics of the response are obtained.Fig.1shows the cumuiative distribution functions obtained from Monte Cario simuiation and SRSM with two K-L varia-bies.A good agreement is observed between thethirdFig.2 Comparison of cumuiative distribution functionsfor SRSM and biack box SSFEMorder stochastic response surface and Monte Cario simuiation（5000sampies）.Thus it can be conciu-ded that SRSM can achieve satisfactory resuits just as anaiyticai SSFEM，without the significant program-ming effort to change the existing finite eiement code. The resuits from third order SRSM（17sampies）and biack box SSFEM（100sampies）are shown in Fig.2.It is shown that SRSM can achieve better resuit u-sing iess sampies than biack box SSFEM with Latin hypercube sampiing.It shouid be noted that K-L expansion with two terms is not enough for an accurate representation of the input random fieid（even for highiy correiated fieid，e.g.!"#=1）.According to the detaiied con-vergence study by Huang［5］，10terms or more wouid be necessary for an accurate representation，which wiii resuit in iarge number of FE runs.SSFEM has the same probiem in the sense that the computationai effort increases dramaticaiiy with the number of terms in K-L expansion.As the same K-L expansion is used for Monte Cario simuiation and SRSM，the inaccuracy in input random fieid does not shown in the resuits. As the purpose of this paper is to show SRSM can work as weii as Monte Cario simuiation with much iess FE runs，the inaccuracy in input random fieid for both methods is not underiined and two-dimension K-L is used for simpie iiiustration.It is noted that the method has been appiied to other more compiicated exampies.The simpie beam probiem is used in this paper for iiiustration purpose.971第2期黄淑萍：基于配点法的谱随机有限元分析———随机响应面法收稿日期：2005-03-18；修改稿收到日期：2005-08-26.基金项目：国家自然科学基金（10602036）资助项目.作者简介：黄淑萍（1973-），女，博士，副教授.5 ConclusionThis paper presented a modified spectrai stochas-tic finite eiement method for probiems in which physi-cai properties exhibit spatiai random variation ，by u-sing the response surface and coiiocation concepts.As regards efficiency ，severai runs or tens of runs in the proposed SRSM can compute the converged soiution statistics.The formaiism of the proposed method is simiiar to the existing SSFEM.However ，in SRSM ，the coefficients in the poiynomiai chaos expansion are caicuiated using a probabiiistic coiiocation approach ，which heips to decoupie the finite eiement and sto-chastic computations.As a resuit ，the finite eiement code can be treated as a biack box ，as in the case of a commerciai code.The coiiocation points in the pro-posed method are optimai for minimizing the mean sguare error and are from high probabiiity regions ，thus ieading to fewer function evaiuations for high pared to the anaiyticai version of SSFEM which uses probabiiistic Gaierkin method ，the advan-tageof the proposed SRSM is that the finite eiementcode can be treated as a biack box.The proposed SRSM is aiso found to reguire iess FEM evaiuations than a biack box version of SSFEM for the same accu-racy.References ：［1］ GHANEM R ，SPANOS P D.Stochastic Finite Elements ：A Spectral Approach ［M ］.Springer -veriag ，1991.［2］ FARAvELLI L.Response surface approach for reiiabiii-ty anaiysis ［J ］.Journal of Engineering Mechanics ，1989，115（12）：763-2781.［3］ GHIOCEL D ，GHANEM R.Stochastic finite eiem-entanaiysis of seismic soii-structure interaction ［J ］.Jour-nalofEngineeringMechanics ，ASCE2002，128（1）：66-77.［4］ ISUKAPALLI S S ，ROY A ，GEORGOPULOS P G.Sto-chastic response surfacemethods （SRSMs ）for uncer-tainty propagation ：Appiication to environmentai and bi-oiogicai systems ［J ］.Risk Analysis ，1998，18（3）：351-363.［5］ HUANG S P ，OUEK S T ，PHOON K K.Convergencestudy of the truncated Karhunen-Loeve expansion for simuiation of stochastic processes ［J ］.International Journal for Numerical methods in Engineering ，2001，52：1029-1043.基于观点法的谱随机有限元分析———随机响应面法黄淑萍（上海交通大学土木工程系，上海200240）摘要：提出了一种基于配点法的谱随机有限元分析方法-随机响应面法（SRSM ），这种方法与已有的谱随机有限元方法（SSFEM ）类似，都用Karhunen-Loeve 级数扩展式表示输入随机场而计算结果的输出用多项式混沌展式表达。

第11章随机响应分析11.1 动力学环境分类11.2 概述1）随机振动是统计意义下描述的振动，在任何瞬时大小未知，但其大小的概率超过一给定的值。

2）常见的例子如地震引起的地基运动、海洋波浪高度和频率、航天器和高耸建筑物受到的风压力、由于火箭与喷气发动机噪音引起的声波等。

3）MSC/NASTRAN 对随机响应分析是作为频率响应后处理进行的。

输入包括频率响应的输出、用户给定的载荷条件（形式为自相关的谱密度）。

输出为响应功率谱密度、自相关函数、响应的均方值。

4）MSC/NASTRAN 随机分析假设历经性随机过程5）随机动态环境例子11.3 自相关与自谱1)自相关函数注：R j(0)为均方值2)自谱函数Fourier变换为3）均方响应值4）外观频率为N06）例子11.4 各态历经性随机激励下线性系统响应计算1) 线性系统单输入输出关系由频率响应分析得到其中，H ja(ω)为频率响应或输入到输出的传递函数对多输入单输出其矩阵形式为输出自相关谱为其单个输入谱为2）线性系统的多输入输出关系多输入输出谱关系其中，输入互谱矩阵为其谱特性为3）常用特殊情况（1）单输入分析（完全相关输入）（2）不相关多输入11.5 MSC/NASTRAN中随机分析的实现1)如果由频率响应计算结果为H ja(ω),但并不直接计算2)如需要H ja(ω),令F a(ω)=111.5.1RANDPS卡片1)定义随机分析中使用的功率谱密度因子，频率相关形式为2) 格式3) 由情况控制卡RANDOM = SID选取4)自谱密度，J=K, X为大于0的整数，Y为05)TID=0, G(F)=011.5.2 TABRND1卡片1)用表格函数定义功率谱密度函数2)格式3）11.5.3 随机响应输入要求1）执行控制2）情况控制3）模型数据11.5.4 随机响应例子1）例1：单输入随机响应分析（1）问题描述：(a)矩形板如图；(b)基座运动（z方向）功率谱(PSD)表中给出；(c)整个频率范围的常临界阻尼比为0.03；(d)用log-log输入PSD;(e)使用模态求解法(2) 使用具有大质量的模态法（在边界处用REB2单元）确定a)9999点处的位移和加速度功率谱(PSD)b)确定结点33和55的位移功率谱(PSD)(3)输入文件ID SEMINAR, PROB10SOL 111TIME 30CENDTITLE= RANDOM ANALYSIS - BASE EXCITATIONSUBTITLE= USING THE MODAL METHOD WITH LANCZOSECHO= UNSORTEDSPC= 101SET 111= 33, 55, 9999ACCELERATION(SORT2, PHASE)= 111METHOD= 100FREQUENCY= 100SDAMPING= 100RANDOM= 100DLOAD= 100$OUTPUT(XYPLOT)XTGRID= YESYTGRID= YESXBGRID= YESYBGRID= YESYTLOG= YESXTITLE= FREQUENCYYTTITLE= ACCEL RESPONSE BASE, MAGNITUDEYBTITLE= ACCEL RESPONSE AT BASE, PHASEXYPLOT ACCEL RESPONSE / 9999 (T3RM, T3IP)YTTITLE= ACCEL RESPONSE AT TIP CENTER, MAGNITUDEYBTITLE= ACCEL RESPONSE AT TIP CENTER, PHASEXYPLOT ACCEL RESPONSE / 33 (T3RM, T3IP)YTTITLE= ACCEL RESPONSE AT OPPOSITE CORNER, MAGNITUDEYBTITLE= ACCEL RESPONSE AT OPPOSETE CORNER, PHASEXYPLOT ACCEL RESPONSE / 55 (T3RM, T3IP)$$ PLOT OUTPUT IS ONLY MEANS OF VIEWING PSD DATA$XGRID= YESYGRID= YESXLOG= YESYLOG= YESYTITLE= ACCEL P S D AT LOADED CORNER XYPLOT ACCEL PSDF / 9999(T3)YTITLE= ACCEL P S D AT TIP CENTERXYPLOT ACCEL PSDF / 33(T3)YTITLE= ACCEL P S D AT OPPOSITE CORNER XYPLOT ACCEL PSDF / 55(T3)$BEGIN BULKPARAM,COUPMASS,1PARAM,WTMASS,0.00259$INCLUDE ’plate.bdf’$GRID, 9999, , 0., 0., 0.$RBE2, 101, 9999, 12345, 1, 12, 23, 34, 45 $SPC1, 101, 12456, 9999$CONM2, 6000, 9999, , 1.0E8$$MAT1, 1, .1, , .1, .286$$ EIGENVALUE EXTRACTION PARAMETERS$EIGRL, 100 , , 2000.$$ SPECIFY MODAL DAMPING$TABDMP1, 100, CRIT,+, 0., .03, 10., .03, ENDT$$ POINT LOADING AT TIP CENTER$RLOAD2, 100, 600, , , 310$TABLED1, 310,+, 10., 1., 1000., 1., END T$DAREA, 600, 9999, 3, 1.E8$$ SPECIFY FREQUENCY STEPS$FREQ,100,30.FREQ1,100,20.,20.,50FREQ4,100,20.,1000.,.03,5$$ SPECIFY SPECTRAL DENSITY$RANDPS, 100, 1, 1, 1., 0., 111$TABRND1, 111,LOG,LOG+, 20., 0.1, 30., 1., 100., 1., 500., .1,+, 1000., .1, ENDT$ENDDATA（4）部分结果基座的加速度PSD(大小、相位)与频率关系结点55的加速度PSD(大小、相位)与频率关系结点33的加速度PSD(大小、相位)与频率关系例2：多输入随机响应分析问题：输入文件ID SEMINAR, PROB11SOL 111TIME 30CENDTITLE= FREQUENCY RESPONSE WITH PRESSURE AND POINT LOADSSUBTITLE= USING THE MODAL METHOD WITH LANCZOSECHO= UNSORTEDSPC= 1SET 111= 11, 33, 55DISPLACEMENT(PLOT, PHASE)= 111METHOD= 100FREQUENCY= 100SDAMPING= 100RANDOM= 100SUBCASE 1LABEL= PRESSURE LOADDLOAD= 100LOADSET= 100SUBCASE 2LABEL CORNER LOADDLOAD= 200LOADSET= 100$OUTPUT (XYPLOT)$XTGRID= YESYTGRID= YESXBGRID= YESYBGRID= YESYTLOG= YESYBLOG= NOXTITLE= FREQUENCY (HZ)YTTITLE= DISPLACEMENT RESPONSE AT LOADED CORNER, MAGNITUDE YBTITLE= DISPLACEMENT RESPONSE AT LOADED CORNER, PHASE XYPLOT DISP RESPONSE / 11 (T3RM, T3IP)YTTITLE= DISPLACEMENT RESPONSE AT TIP CENTER, MAGNITUDE YBTITLE= DISPLACEMENT RESPONSE AT TIP CENTER, PHASEXYPLOT DISP RESPONSE / 33 (T3RM, T3IP)YTTITLE= DISPLACEMENT RESPONSE AT OPPOSITE CORNER, MAGNITUDE YBTITLE= DISPLACEMENT RESPONSE AT OPPOSITE CORNER, PHASE XYPLOT DISP RESPONSE / 55 (T3RM, T3IP)$$ PLOT OUTPUT IS ONLY MEANS OF VIEWING PSD DATA$XGRID= YESYGRID= YESXLOG= YESYLOG= YESYTITLE= DISP P S D AT LOADED CORNERXYPLOT DISP PSDF / 11(T3)YTITLE= DISP P S D AT TIP CENTERXYPLOT DISP PSDF / 33(T3)YTITLE= DISP P S D AT OPPOSITE CORNERXYPLOT DISP PSDF / 55(T3)$BEGIN BULKPARAM,COUPMASS,1PARAM,WTMASS,0.00259$$ MODEL DESCRIBED IN NORMAL MODES EXAMPLE$INCLUDE ’plate.bdf’$$ EIGENVALUE EXTRACTION PARAMETERS $EIGRL, 100, 10., 2000.$$ SPECIFY MODAL DAMPING$TABDMP1, 100, CRIT,+, 0., .03, 10., .03, ENDT$$ FIRST LOADING$RLOAD2, 100, 300, , , 310$TABLED1, 310,+, 10., 1., 1000., 1., ENDT$$ UNIT PRESSURE LOAD TO PLATE$LSEQ, 100, 300, 400$PLOAD2, 400, 1., 1, THRU, 40$$ SECOND LOADING$RLOAD2, 200, 600, , , 310$$ POINT LOAD AT TIP CENTER$DAREA, 600, 11, 3, 1.$$ SPECIFY FREQUENCY STEPS$FREQ1, 100, 20., 20., 49$$ SPECIFY SPECTRAL DENSITY$RANDPS, 100, 1, 1, 1., 0., 100 RANDPS, 100, 2, 2, 1., 0., 200 RANDPS, 100, 1, 2, 1., 0., 300 RANDPS, 100, 1, 2, 0., 1., 400$TABRND1, 100,+, 20., 0.1, 30., 1., 100., 1., 500., .1,+, 1000., .1, ENDT$TABRND1, 200,+, 20., 0.5, 30., 2.5, 500., 2.5, 1000., 0.,+, ENDT$TABRND1, 300,+, 20., -.099619, 100., -.498097, 500., .070711, 1000., 0., +, ENDT$TABRND1, 400,+, 20., .0078158, 100., .0435791, 500., -.70711, 1000., 0., +, ENDT$ENDDATA。

相关激励作用下随机结构振动响应的统计分析廖庆斌,李舜酩,辛江慧,郑娟丽(南京航空航天大学能源与动力学院,江苏南京210016)摘要:应用随机过程理论,以能量为变量,分析了随机结构振动响应的统计特性。

结构受相关激励作用时,通过输入激励的解相关方法,将作用在结构上的相关激励转变为各个不相关激励的作用;分析结构的振动响应的统计特性时,计及响应特征频率的相关性,在响应特征频率满足高斯正交总体的假设下,推导出了随机结构振动响应分析的统计分析表达式。

应用设计的实验件和试验验证了所提出的统计分析的正确性,通过和已存在的统计分析结果的比较,表明了统计分析具有更高的分析精度,能够定性和定量的给出随机结构振动响应的统计变化情况。

关键词:随机结构;相关激励;统计分析;本征正交分解;统计能量分析中图分类号:T B53;O324 文献标识码:A 文章编号:1004-4523(2008)05-0429-07引　言结构的动力响应特性与激励频率有很大的关系,在激励频率较低时,结构只有很少的前几阶模态被激起,这样应用有限元或者边界元方法即可以精确地得到系统动态响应,当激励频率较高(中频或者高频)时,结构的模态被大量的激起,此时要准确地计算其振动响应变得非常困难[1]。

解决中、高频振动的有效方法是Lyon等人提出的统计能量分析(Statistical Energy Analysis:SEA)方法[2],他将随机动力系统划分为数量不多的动力子结构,然后求解各个子系统的振动能量,进而得到动力系统的振动响应。

在分析系统的中、高频振动响应时,SEA方法包含有振动能量的平均分布、系统响应的频带平均以及系统响应的随机总体平均等假设[1,3],因此, SEA方法仅仅是结构动力响应的估计。

Kompella 和Bernhar d等人通过实验发现[4],由同一条生产线生产出来的98辆型号相同的汽车,对其进行响应分析(振动和噪声水平分析)时,车辆的动态响应敏感的依赖于制造细节的变化。